The document discusses the definitions, history, and applications of eigenvalues and eigenvectors. It defines eigenvalues as scalars that satisfy the equation Ax = λx for a matrix A, and eigenvectors as non-zero vectors that satisfy this same equation. The document traces the history of eigenvalues and eigenvectors from their discovery and study in the 18th-19th centuries to modern applications. It provides examples of applications in areas like communication systems, structural analysis, and oil exploration.

1. IN THE NAME OF
ALLAH
THE RAHMAN ,
THE RAHIM
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2. EIGENVECTOR AND
EIGENVALUES
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3. INTRODUCTION:
• Eigenvalues and eigenvectors feature prominently in the analysis of linear transformations.
• Eigen is a German word meaning “proper” or charachteristics.
• It is used to study the principle axis of rotational motion of rigid bodies.
• There are many applications of eigenvectors and eigenvalues one of them is matrix diagonalization.
• Eigenvalues are often introduced in the context of linear algebra or matrix theory.
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4. HISTORY :
• In the 18th century Euler studied the rotational motion of a rigid body and discovered the
importance of the principal axes.
• Joseph-Louis Lagrange realized that the principal axes are the eigenvectors of the inertia
matrix.
• In 19th century Cauchy used there work to classify quadric surface and name this term as
characteristic root and now they are called as eigenvalues.
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5. • Liouville studied eigenvalue problems the discipline that grew out of their work is now called Sturm–
Liouville theory.
• At the start of the 20th century, Hilbert studied the eigenvalues of integral operators by viewing the
operators as infinite matrices. He was the first to use the German word eigen, which means "own", to
denote eigenvalues and eigenvectors in 1904.
• The first algorithm for computing eigenvector and values was appeared in 1929.
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6. DEFINITION:
A scalar λ is called an eigenvalue of the n × n matrix A if there is a non zero x of Ax = λx. Such an x is called
an eigenvector corresponding to the eigenvalue λ. Suppose that A is matrix for a linear transformation T : Rn
→ Rn.
• An eigenvalue of a square matrix is a scalar that is usually represented by the Greek letter λ (pronounced
lambda).
• As you might suspect, an eigenvector is a vector. Moreover, we require that an eigenvector be a non-zero
vector, in other words, an eigenvector can not be the zero vector.
• We will denote an eigenvector by the small letter x.
• All eigenvalues and eigenvectors satisfy the equation for a given square matrix.
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7.  Linear algebra studies linear transformation ,which are represented by
matrices acting on vectors.
 Eigenvalues, eigenvectors and Eigen spaces are properties of a matrix.
 In general, a matrix acts on a vector by changing both its magnitude
and its direction. However, a matrix may act on certain vectors by
changing only their magnitude, and leaving their direction unchanged or
possibly reversing it.
 These vectors are the eigenvectors of the matrix. A matrix acts on an
eigenvector by multiplying its magnitude by a factor, which is positive if
its direction is unchanged and negative if its direction is reversed.
 This factor is the eigenvalue associated with that Eigen vector.
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8.  Eigenvectors are vectors that point in directions where there is no rotation.
 Eigenvalues are the change in length of the eigenvector from the original length.
 The basic equation in eigenvalue problems Ax=λx
 To find the eigenvalues of an n × n matrix A . We rewrite Ax= λx as Ax= λIx
 or equivalently, (-λI +A)x=0
 First has a nonzero solution if and only if det (-λI +A)=0
 Second is called the characteristic equation of A; the scalar satisfying this
equation are the eigenvalues of A.
 When expanded ,det (-λI + A) is a polynomial p in λ called the characteristic
polynomial of A.
 The set of all eigenvalues of A is called the Spectrum of A.
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9. Eigen Values of Triangular Matrix are equal to Entries on its main diagonal.
Types of Matrices Nature of Eigenvalues
Symmetric Reals
Skew Symmetric Purely Imaginary or Zero
Orthogonal Unit modulus
Hermitian Reals
Skew Hermitian Purely Imaginary or Zero
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10.  If k is a positive integer and λ is an eigenvalue of a matrix A
and x is corresponding eigenvector, then λ raise to power k
is an Eigen value of A raise to power k and x is a corresponding
Eigen vector.
 A square matrix A is invertible if and only if λ=0 is not an
Eigen value of A.
 Dimension of Eigen space of n*n matrix are less than are
equal to number of Eigen values.
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11.  Diagonalization
 Definition :
A square matrix A is called diagonalizable if there is an invertible
matrix P such that P-1 A P is a diagonal matrix, than the matrix P
is said to diagonalize A.
 If an n × n matrix A has n distinct eigenvalues, then A
is
diagonalizable.
 Mean sum of dimensions of distinct Eigen spaces is equal to order
of original matrix.
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12.  Eigenvectors and eigenvalues are used in structural geology to determine the directions of principal strain
the directions where angles are not changing.
 In seismology, these are the directions of least compression (tension), the compression axis, and the
intermediate axis (in three dimensions).
 Eigenvectors and eigenvalues have many applications, particularly in physics. Consider rigid
physical bodies. Rigid physical bodies have a preferred direction of rotation, about which they can
rotate freely.
 For example, if someone were to throw a football it would rotate around its axis while flying
prettily through the air. If someone were to hit the ball in the air, the ball would be likely to flop
in a less simple way.
 Although this may seem like common sense, even rigid bodies with a more complicated shape will
have preferred directions of rotation. These are called axes of inertia, and they are calculated by
finding the eigenvectors of a matrix called the inertia tensor.
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13.  The product of the eigenvalues = det|A|•
 The sum of the eigenvalues = trace(A)
 If (p-1 A p)=B then A and Bare called similar matrices.
 There are some properties of similar matrices given below
 Det (A)= Det (B)
 Rank (A) = Rank (B)
 If A is invertible than so is B
 Eigen values & Dimensions of eigen spaces are same.
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14. APPLICATIONS OF EIGEN VALUES AND EIGEN VECTORS
Communication systems:
Eigenvalues were used by Claude Shannon to determine the
theoretical limit to
how much information can be transmitted through a
communication medium
like your telephone line or through the air. This is done by
calculating the
eigenvectors and eigenvalues of the communication channel
(expressed a
matrix), and then water filling on the eigenvalues. The eigenvalues
are then, in
essence, the gains of the fundamental modes of the channel, which
themselves are captured by the eigenvectors.
Designing bridges:
The natural frequency of the bridge is the eigenvalue of smallest
magnitude of
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15. Designing car stereo system:
Eigenvalue analysis is also used in the design of the
car stereo
systems, where it helps to reproduce the vibration of
the car due
to the music.
Mechanical Engineering:
Eigenvalues and eigenvectors allow us to "reduce" a
linear
operation to separate, simpler, problems. For example,
if a stress is
applied to a "plastic" solid, the deformation can be
dissected into
"principle directions"- those directions in which the
deformation is
greatest. Vectors in the principle directions are the
eigenvectors
and the percentage deformation in each principle
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16. SOME OTHER APPLICATIONS:
Eigenvalues are not only used to explain natural occurrences, but also to
discover new and better designs for the future. Some of the results are quite
surprising. If you were asked to build the strongest column that you could to
support the weight of a roof using only a specified amount of material, what
shape would that column take? Most of us would build a cylinder like most
other columns that we have seen. However, Steve Cox of Rice University and
Michael Overton of New York University proved, based on the work of J.
Keller and I. TadJ bakhsh, that the column would be stronger if it was largest
at the top, middle, and bottom. At the points of the way from either end, the
column could be smaller because the column would not naturally buckle there
anyway. Does that surprise you? This new design was discovered through the
study of the eigenvalues of the system involving the column and the weight
from above. Note that this column would not be the strongest design if any
significant pressure came from the side, but when a column supports a roof,
the vast majority of the pressure comes directly from above.
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17. Oil companies
 Oil companies frequently use eigenvalue analysis to explore land for
oil.
 Oil , dirt, and other substances all give rise to linear systems
which have different eigenvalues, so eigenvalue analysis can give a
good indication of where oil reserves are located.
 Oil companies place probes around a site to pick up the waves
that result
from a huge truck used to vibrate the ground.
 The waves are changed as they pass through the different
substances in
the ground. The analysis of these waves directs the oil
companies to
possible drilling sites.
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18. Applications
Vibration analysis
Civil Engineering
Google search
Sharing information
Face recognition
Musical instruments
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